Graph Regularized Non-negative Matrix Factorization By Maximizing Correntropy

Non-negative matrix factorization (NMF) has proved effective in many clustering and classification tasks. The classic ways to measure the errors between the original and the reconstructed matrix are $l_2$ distance or Kullback-Leibler (KL) divergence. However, nonlinear cases are not properly handled when we use these error measures. As a consequence, alternative measures based on nonlinear kernels, such as correntropy, are proposed. However, the current correntropy-based NMF only targets on the low-level features without considering the intrinsic geometrical distribution of data. In this paper, we propose a new NMF algorithm that preserves local invariance by adding graph regularization into the process of max-correntropy-based matrix factorization. Meanwhile, each feature can learn corresponding kernel from the data. The experiment results of Caltech101 and Caltech256 show the benefits of such combination against other NMF algorithms for the unsupervised image clustering

LASS: a simple assignment model with Laplacian smoothing

We consider the problem of learning soft assignments of $N$ items to $K$ categories given two sources of information: an item-category similarity matrix, which encourages items to be assigned to categories they are similar to (and to not be assigned to categories they are dissimilar to), and an item-item similarity matrix, which encourages similar items to have similar assignments. We propose a simple quadratic programming model that captures this intuition. We give necessary conditions for its solution to be unique, define an out-of-sample mapping, and derive a simple, effective training algorithm based on the alternating direction method of multipliers. The model predicts reasonable assignments from even a few similarity values, and can be seen as a generalization of semisupervised learning. It is particularly useful when items naturally belong to multiple categories, as for example when annotating documents with keywords or pictures with tags, with partially tagged items, or when the categories have complex interrelations (e.g. hierarchical) that are unknown.Comment: 20 pages, 4 figures. A shorter version appears in AAAI 201